Polynomial Subspace Dimension & Basis Calculation

[gotmejerry]

Homework Statement

Let [itex]M[/itex] be a subspace of the vector space [itex]\mathbb{R}_2[t] [/itex] generated by [itex]p_1(T)=t^2+t+1[/itex] and [itex]p_2(T)=1-t^2[/itex], and [itex]N[/itex] be a subspace generated by [itex]q_1(T)=t^2+2t+3[/itex] and [itex]q_2(T)=t^2-t+1[/itex]. Show the dimension of the following subspaces:[itex] M+N[/itex], [itex]M \cap N[/itex], and give a basis for each.

Homework Equations

The Attempt at a Solution

I have tried the following: if I take the linear combination of [itex]p_1[/itex] [itex]p_2[/itex] [itex]q_1[/itex] [itex]q_2[/itex], I get [itex](a+b+c+d)t^2 + (a+2c-d)t +(a+b+3c+d).[/itex] And a basis of this polynomial is [itex]\{1,t,t^2\}[/itex], which means the dimension of M+N is 3.

And if M and N are finite dimension subspaces then [itex]dim(M+N)=dim M + dim N- dim(M \cap N)[/itex]. The diemnsion of the subspace generated by p1 and p2 is 2, and so is the dimension of the subspace generated by q1 and q2. Am I right? But then from this [itex]dim(M+N)=dim M + dim N- dim(M \cap N)[/itex] I get that [itex](M \cap N)[/itex] has a dimension of 1.

Thank you!

 

[Deveno]

1. how do you conclude that dim(M+N) = 3? clearly it's ≤ 3, but how do you know that we need all 3 basis elements to describe something in M+N? it's not immediately clear that span({p₁,p₂,q₁,q₂}) is all of R₂[t].

2. before you can conclude dim(M) and dim(N) = 2, you should show that {p₁,p₂} and {q₁,q₂} are actually linearly independent. this isn't hard to do.

3. assuming you do (1) and (2) above, then you are in a position to use

dim(M+N) = dim(M) + dim(N) - dim(M∩N).

finding the dimensions isn't really the hard part. finding the basis elements is. what would an element of M∩N look like?

 

[gotmejerry]

1. If I take the linear combinations of p1 p2 q1 q2 as I have written and I am not wrong I think the basis {t^2 t 1} is ok.

2. I found out that {p1,p2} and {q1,q2} are linearly independent, because a*p1+b*p2=0 s only solution is the trivial soulution, same for {q1,q2}. So dim(M)=2 and dim(N)=2.

3. An element in [itex]M \cap N[/itex] is something which is in the span({p1,p2}) and in the span({q1,q2}) at the same time. But how do I find a basis ?

 

[Deveno]

for span({p₁,p₂,q₁,q₂}) to equal R₂[t], every polynomial must be in the span.

if we have any abitrary polynomial At² + Bt + C, in the span, this means that we have:

a+b+c+d = A
a+2c-d = B
a+b+3c+d = C

this is equivalent to:

[tex]\begin{bmatrix}1&1&1&1\\1&0&2&-1\\1&1&3&1 \end{bmatrix}\begin{bmatrix}a\\b\\c\\d \end{bmatrix} = \begin{bmatrix}A\\B\\C \end{bmatrix}[/tex]

now, it isn't obvious that the 4x3 matrix actually has rank 3. i think you might have made a lucky guess.

as for M∩N, suppose we have a(1+t+t²) + b(1-t²) = c(1-t+t²) + d(3+2t+t²). what can we say about a,b,c and d?

 

[gotmejerry]

a(1+t+t2) + b(1-t2) = c(1-t+t2) + d(3+2t+t2) is true if

a-b=c+d
a=2d-c
a+b=c+3d

What do I do now?